Non-invertible

A mathematical term describing a function, transformation, or matrix that does not possess an inverse. In other words, applying the original operation to a result does not guarantee returning to the original starting point. This often indicates a loss of information or a reduction in dimensionality. Non-invertibility arises when an operation maps multiple inputs to the same output or when the operation alters the number of dimensions of the space. It's a fundamental concept in various fields like linear algebra, cryptography, and signal processing, limiting the reversibility of processes and having far-reaching consequences for information recovery and system analysis. Such an item cannot be undone or reversed because of this lack of an inverse or reversed operation.

Non-invertible meaning with examples

• In linear algebra, a square matrix is non-invertible (also known as singular or degenerate) if its determinant is zero. This means the matrix does not have a multiplicative inverse, making the system of linear equations unsolvable or possessing infinitely many solutions. non-invertible matrices lead to data loss in transformations.
• Consider a function f(x) = x². Since both positive and negative values of x can produce the same output, such as f(2) = 4 and f(-2) = 4, this quadratic function is non-invertible over the real numbers because one cannot uniquely determine the original input from the output.
• In cryptography, some encryption algorithms might be intentionally designed to be non-invertible, or at least computationally difficult to invert, without the correct decryption key. This irreversibility is essential to protect sensitive information and prevent unauthorized access to the encrypted data.
• A compression algorithm that discards information to reduce the size of a file can result in non-invertible transformations. Reversing the process without the original, uncompressed data will lead to an approximate solution only, as information is lost, and is, therefore, non-invertible.
• A projection, like projecting a 3D object onto a 2D plane, is typically a non-invertible transformation. Information about the third dimension is lost, and it's impossible to reconstruct the original 3D object from the 2D projection alone, unless information of the depth is available.